Real-rooted atlas

A225470 → basis A013610 → D+1

nonnegative coefficients

Proved for all n all roots real, in (−∞, 0]coefficients of one signHadamard kernelinterlacing (weak)

Recurrence

\(P_{n}(x) = \left(-9 + 18\,x - 9\,x^{2} + 18\,n - 18\,n x - 9\,n^{2}\right)\,P_{n-2}(x) + \left(-3 + 6\,x + 6\,n\right)\,P_{n-1}(x)\)
\(P_{0}(x) = 1\)
\(P_{1}(x) = 6 + 3\,x\)

Rows ?

nx0x1x2x3x4x5x6x7x8x9x10x11x12x13x14Pn(1)
011
1639
24545999
3459621243271350
459949720526511348122194
595742175932116640352354860243428652
61810836365374827913411046115204120196837299526572
73960219686019084731310933194988378075901071630765452187239660208
898352014422690037522099362536103060188029609136951254532523567828868465616735365136
92734866244866370603608658148344203544868934011528136270236142855930587382024210090106288219683209203521120
108418875745602133782381808224280360572413036377637804680935280451091634729571681851332716952967901071739353838185590497117993208160
1128425120994080748233130134248267312386198851238174718412199632986155355166471528801910450372293109615552101885646426545792499513640319177147263318896863360
1210445177798956802842976570338944328090639815525621485321186163128968372910614262529720467033764967733445002968674602290586650415473134407771654018999015754782969053144110523998066976640
1341499876683848320116379509654676096139563809058695112958898682787827604247005068163657812884964074374392276806114371987242790624418213147733887130930380307098070921066108663676894198291658095921594323451935801295937280
14177277846817038464051071550181507307526338029782609842256454322594621678328021190391218412161206845351959023898321586965168409190572692212095103745533682724255852103099830012329052069266401001897278415006530467512530569173311478296920754082122599750400
Technical details
Node id
n-9db4cace849774b0 · revision 0 · JSON
Definition source
derived:one_plus_derivative
Fingerprint
f6-9dfcd5dbd8ed9f8f

Observed on the stored rows ?

  • Degrees: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 (affine: slope 1, intercept 0)
  • Palindromic rows: no
  • γ-positive (gamma_positive): no
  • Nonnegative coefficients (nonn): yes
  • Only rational roots (rr_by_factors: every nonzero row is a product of linear factors over ℚ): no

Matches ?

No OEIS matches.

Submission text ?

Atlas rows n = 0, …, ascending powers of x
%e Example
Triangle begins:
  0:           1;
  1:           6,           3;
  2:          45,          45,           9;
  3:         459,         621,         243,          27;
  4:        5994,        9720,        5265,        1134,          81;
  5:       95742,      175932,      116640,       35235,        4860,        243;
  6:     1810836,     3653748,     2791341,     1046115,      204120,      19683,       729;
  7:    39602196,    86019084,    73131093,    31949883,     7807590,    1071630,     76545,     2187;
  8:   983520144,  2269003752,  2099362536,  1030601880,   296091369,   51254532,   5235678,   288684,    6561;
  9: 27348662448, 66370603608, 65814834420, 35448689340, 11528136270, 2361428559, 305873820, 24210090, 1062882, 19683;
%F Formula
The row polynomials satisfy P(n,x) = -(9*n^2-18*n+9+(18*n-18)*x+9*x^2)*P(n-2,x) + (6*n-3+6*x)*P(n-1,x) for n >= 2, with P(0,x) = 1, P(1,x) = 6 + 3*x.
%t Mathematica
P[0, x_] := 1;
P[1, x_] := 6 + 3*x;
P[n_, x_] := P[n, x] = Expand[-(9*n^2 - 18*n + 9 + (18*n - 18)*x + 9*x^2)*P[n - 2, x] + (6*n - 3 + 6*x)*P[n - 1, x]];
Table[CoefficientList[P[n, x], x], {n, 0, 9}] // Flatten
%o PARI/GP
P(N) = {my(v = vector(N + 1)); v[1] = 1; v[2] = 6 + 3*x; for(n = 2, N, my(i = n + 1); v[i] = -(9*n^2 - 18*n + 9 + (18*n - 18)*x + 9*x^2)*v[i - 2] + (6*n - 3 + 6*x)*v[i - 1]); v};
row(n, p) = {my(c = Vecrev(p)); c};
T(N) = {my(v = P(N)); concat(vector(#v, i, row(i - 1, v[i])))};
T(9)

polytool c48629374597

Properties

Proof chains ? JSON

Same fingerprint ?

References

  1. [Bra06] Petter Brändén. On linear transformations preserving the Pólya frequency property. Transactions of the American Mathematical Society, 358(8):3697–3716, February 2006. doi:10.1090/s0002-9947-06-03856-6
  2. [Bra15] Petter Brändén. Unimodality, log-concavity, real-rootedness and beyond. Handbook of enumerative combinatorics, :437–483, March 2015. doi:10.1201/b18255-10
  3. [LW07] Lily L. Liu and Yi Wang. A unified approach to polynomial sequences with only real zeros. Advances in Applied Mathematics, 38(4):542–560, May 2007. doi:10.1016/j.aam.2006.02.003
  4. [SP14] J Schur and G Polya. Über zwei Arten von Faktorenfolgen in der Theorie der algebraischen Gleichungen. J. Reine Angew. Math., 1914.